Dirac cohomology, unitary representations and a proof of a conjecture of Vogan

Abstract

Let G G be a connected semisimple Lie group with finite center. Let K K be the maximal compact subgroup of G G corresponding to a fixed Cartan involution θ \theta . We prove a conjecture of Vogan which says that if the Dirac cohomology of an irreducible unitary ( g , K ) (\mathfrak {g},K) -module X X contains a K K -type with highest weight γ \gamma , then X X has infinitesimal character γ + ρ c \gamma +\rho _{c} . Here ρ c \rho _{c} is the half sum of the compact positive roots. As an application of the main result we classify irreducible unitary ( g , K ) (\mathfrak {g},K) -modules X X with non-zero Dirac cohomology, provided X X has a strongly regular infinitesimal character. We also mention a generalization to the setting of Kostant’s cubic Dirac operator.